Related papers: Szlenk indices of convex hulls

The notion of ball convexity, considered in finite dimensional real Banach spaces, is a natural and useful extension of usual convexity; one replaces intersections of half-spaces by suitable intersections of balls. A subset $S$ of a normed…

Let $\mu$ be a probability measure on a separable Banach space $X$. A subset $U\subset X$ is $\mu$-continuous if $\mu(\partial U)=0$. In the paper the $\mu$-continuity and uniform $\mu$-continuity of convex bodies in $X$, especially of…

We show that for a $\sigma $-ideal $\ci$ with a Steinhaus property defined on Banach space, if two non-homeomorphic Banach with the same cardinality of the Hamel basis then there is a $\ci$ nonmeasurable subset as image by any isomorphism…

The relation between different notions measuring proximity to $\ell_1$ and distortability of a Banach space is studied. The main result states that a Banach space, whose all subspaces have Bourgain $\ell_1$ index greater than…

A subset $A$ of a Banach space is called Banach-Saks when every sequence in $A$ has a Ces{\`a}ro convergent subsequence. Our interest here focusses on the following problem: is the convex hull of a Banach-Saks set again Banach-Saks? By…

We study an infinite system of ordinary differential equations that models the evolution of coagulating and fragmenting clusters, which we assume to be composed of identical units. Under very mild assumptions on the coefficients we prove…

Being motivated by the famous Kaplansky theorem we study various sequential properties of a Banach space $E$ and its closed unit ball $B$, both endowed with the weak topology of $E$. We show that $B$ has the Pytkeev property if and only if…

Using the game approach to fragmentability, we give new and simpler proofs of the following known results: (a) If the Banach space admits an equivalent Kadec norm, then its weak topology is fragmented by a metric which is stronger than the…

Quasi-invariant and pseudo-differentiable measures on a Banach space $X$ over a non-Archimedean locally compact infinite field with a non-trivial valuation are defined and constructed. Measures are considered with values in $\bf R$.…

In this paper the theory of uniformly convex metric spaces is developed. These spaces exhibit a generalized convexity of the metric from a fixed point. Using a (nearly) uniform convexity property a simple proof of reflexivity is presented…

We introduce slicely countably determined points (SCD points) of a bounded and convex subset of a Banach space which extends the notions of denting points, strongly regular points and much more. We completely characterize SCD points in the…

We construct a locally finite graph and a bounded geometry metric space which do not admit a quasi-isometric embedding into any uniformly convex Banach space. Connections with the geometry of $c_0$ and superreflexivity are discussed.

In complete metric measure spaces equipped with a doubling measure and supporting a weak Poincar\'e inequality, we investigate when a given Banach-valued Sobolev function defined on a subset satisfying a measure-density condition is the…

We show that for real Banach spaces that are either separable or dual spaces, the Lipschitz numerical index coincides with the classical (linear) numerical index. This result provides partial evidence toward the question posed by Wang,…

the main goal of this paper is to prove that any Banach space X, that every dual ball in X** is weak* -separable, or every weak* -closed convex subset in X** is weak* -separable, or every norm-closed convex set in X* is constructible,…

We introduce extensions of $\Delta$-points and Daugavet points in which slices are replaced by relative weakly open subsets (super $\Delta$-points and super Daugavet points) or by convex combinations of slices (ccs $\Delta$-points and ccs…

Inspired by R. Whitley's thickness index the last named author recently introduced the Daugavet index of thickness of Banach spaces. We continue the investigation of the behavior of this index and also consider two new versions of the…

A subset of a Banach space is called equilateral if the distances between any two of its distinct elements are the same. It is proved that there exist non-separable Banach spaces (in fact of density continuum) with no infinite equilateral…

We give a new scale of completeness conditions for exponential systems in two types of functional spaces on subsets of the complex plane. The first is the Banach spaces of functions that are continuous on a compact and simultaneously…

Contraction rates of time-varying maps induced by dynamical systems illuminate a wide range of asymptotic properties with applications in stability analysis and control theory. In finite-dimensional smoothly varying inner-product spaces…